ELECTRIC CIRCUITS EIGHTH EDITION JAMES W. NILSSON & SUSAN A. RIEDEL

CHAPTER 11 BALANCED THREE–PHASE CIRCUITS © 2008 Pearson Education

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CONTENTS 11.1 Balanced Three-Phase Voltages 11.2 Three-Phase Voltage Sources 11.3 Analysis of the Wye-Wye Circuit 11.4 Analysis of the Wye-Delta Circuit 11.5 Power Calculations in Balanced Three- Phase Circuits 11.6 Measuring Average Power in Three- Phase Circuits © 2008 Pearson Education

5 Constant rms vtg. must be supplied whether lightly loaded, as at 3:00 am, or heavily loaded, as at midafternoon on a hot, humid day. One technique for maintaining vtg. levels on a utility sys. is to place capacitors at strategic locations in the distribution network.

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11.1 Balanced Three-Phase Voltages A basic three-phase circuit © 2008 Pearson Education

3 vtgs.: A-phase vtg. B-phase vtg. C-phase vtg.

11.1 Balanced Three-Phase Voltages Phasor diagrams of a balanced set of three-phase voltages © 2008 Pearson Education The abc (positive) sequence The abc (negative) sequence

11.2 Three-Phase Voltage Sources © 2008 Pearson Education a three-phase voltage source A three-phase voltage source is a generator with three separate windings distributed around the periphery of the stator. Rotor: electromagnet driven by steam or gas turbine. Stator is the stationary part of a rotor system and is winded by 3 seperated windings. rotor

  The two basic connections of an ideal three- phase source. © 2008 Pearson Education 11.2 Three-Phase Voltage Sources A Y-connected source A ∆-connected source

A model of a three-phase source with winding impedance: (a) A Y-connected source (b) A ∆-connected source © 2008 Pearson Education 11.2 Three-Phase Voltage Sources

Basic ckt configuration between S. and load

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11.3 Analysis of the Wye-Wye Circuit © 2008 Pearson Education : internal impedance of each phase in generator : line impedance : load impedance

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See: = 0By the way, Therefore, (balanced 3-phase vtg.) Also,

Now turn to the effect that balanced conds. have 3 line ct.

11.3 Analysis of the Wye-Wye Circuit A single-phase equivalent circuit © 2008 Pearson Education  A single-phase equivalent circuit is used to calculate the line current and the phase voltage in one phase of the Y-Y structure. The a-phase is normally chosen for this purpose.

Caution: ct. in neutral conductor is I aA, which is not the same as ct. in neutral conductor, that is, (I aA,or I bB,or I cC ), We can find any vtg. easily.

 Phasor diagrams showing the relationship between line-to-line and line-to-neutral voltages in a balanced system. © 2008 Pearson Education 11.3 Analysis of the Wye-Wye Circuit The abc sequenceThe acb sequence

Above expressions explain the following:

Terminology: Line vtg. (V L ) refers to vtg. across any pair of lines. Phase vtg. (V Φ ) refers to vtg. across a single phase. Line ct. (I L ) refers to ct. in a single line. Phase ct.(I Φ ) refers to ct. in a single phase. See: Line vtg. = phase vtg. Line ct. = phase ct.

Balanced 3 phase Y-connected generator: - impedance of gen.:(0.2 + j0.5)Ω - internal vtg. :120 V/φ Balanced 3 phase Y-connected impedance: (39 + j28)Ω/φ Line impedance: (0.8 + j1.5)Ω a phase vtg. : reference phasor vtg.

30 After the Δ load has been replaced by its Y equivalent, the a-phase can be modeled by the single phase equivalent circuit shown in Fig

11.4 Analysis of the Wye-Delta Circuit A circuit used to establish the relationship between line currents and phase currents in a balanced ∆ load © 2008 Pearson Education

 Phasor diagrams showing the relationship between line currents and phase currents in a ∆-connected load. © 2008 Pearson Education 11.4 Analysis of the Wye-Delta Circuit The positive sequence The negative sequence

Y-connected S.: - impedance of gen.:(0.2 + j0.5)Ω - internal vtg. :120 V/φ Δ-connected load impedance: ( j85.8)Ω/φ cf: Y-connected impedance: (39 + j28)Ω/φ Line impedance: (0.3 + j0.9)Ω a phase vtg. : reference phasor vtg. Y-to-Y ckt

Line vtg. = phase vtg. Line ct. = phase ct. See:

11.5 Power Calculations in Balanced Three-Phase Circuits A balanced Y load used to introduce average power calculations in three-phase circuits © 2008 Pearson Education  Average Power in a Balanced Wye Load

See: - all phasor ct. & vtg. are rms value. - in this sys.,

11.5 Power Calculations in Balanced Three-Phase Circuits Total real power in a balanced three-phase load © 2008 Pearson Education

11.5 Power Calculations in Balanced Three-Phase Circuits Total reactive power in a balanced three-phase load © 2008 Pearson Education  Complex Power in a Balanced Wye Load

Total complex power in a balanced three-phase load 11.5 Power Calculations in Balanced Three-Phase Circuits

Δ-connected load used to discuss power calculations © 2008 Pearson Education 11.5 Power Calculations in Balanced Three-Phase Circuits  Power Calculations in a Balanced Delta Load

Note: in a balanced load, av. power per phase is equal in Y- and Δ-connected. Reactive power & complex power are also same.

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Check! V s =

Y-to-Y ckt

11.6 Measuring Average Power in Three-Phase Circuits © 2008 Pearson Education A wattmeter measures the average power delivered to a load by using a current coil connected in series with the load and a potential coil connected in parallel with the load. : stationary & designed to carry a ct. proportional to the load ct. : movable & carries a ct. proportional to the load vtg.

11.6 Measuring Average Power in Three-Phase Circuits © 2008 Pearson Education The key features of the electrodynamometer wattmeter

© 2008 Pearson Education A general circuit whose power is supplied by n conductors

 The total average power in a balanced three-phase circuit can be measured by summing the readings of two wattmeters connected in two different phases of the circuit. © 2008 Pearson Education 11.6 Measuring Average Power in Three-Phase Circuits

63 If we wish to measure the total power at the terminals of the box, we need to know n−1 currents and voltages. Thus the total power is the sum of n − 1 product terms; that is

64 Only two wattmeters are needed to measure the total average power in any balanced three-phase system.

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68 Calculate the reading of each wattmeter. Where phase vtg. At the load = 120 V a) Zφ = 8 + j6Ω b) Zφ = 8 - j6Ω c) Zφ = 5 + j5√3Ω d) Zφ = 10 ∠ -75 o Ω

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